To determine the temperature rise in a closed vessel containing water when a paddle wheel is connected to a shaft of an engine, we need to consider the principles of thermodynamics and the work done by the paddle wheel on the water. Let’s break this down step by step.
Understanding the System
In this scenario, we have a closed vessel filled with water, and a paddle wheel is attached to a shaft. When the engine turns the shaft, the paddle wheel rotates within the water, doing work on it. This work results in an increase in the internal energy of the water, which manifests as a rise in temperature.
Key Concepts
- Work Done: The work done by the paddle wheel on the water can be calculated using the formula: Work = Torque × Angular Displacement.
- First Law of Thermodynamics: This law states that the change in internal energy (ΔU) of a system is equal to the heat added to the system (Q) minus the work done by the system (W): ΔU = Q - W.
- Specific Heat Capacity: The amount of heat required to raise the temperature of a unit mass of a substance by one degree Celsius is known as specific heat capacity (c). For water, this value is approximately 4.18 J/g°C.
Calculating Temperature Rise
To find the temperature rise (ΔT) of the water, we can use the relationship between the work done on the water and the heat absorbed by it. The work done on the water by the paddle wheel will increase its internal energy, leading to a temperature rise. The formula we can use is:
Q = mcΔT
Where:
- Q = heat energy absorbed by the water (in joules)
- m = mass of the water (in grams)
- c = specific heat capacity of water (4.18 J/g°C)
- ΔT = change in temperature (in °C)
Steps to Calculate ΔT
- Calculate the work done by the paddle wheel using the torque and angular displacement.
- Assume that all the work done is converted into heat energy absorbed by the water (Q = Work done).
- Rearrange the specific heat formula to solve for ΔT: ΔT = Q / (mc).
- Substitute the values for Q, m, and c into the equation to find the temperature rise.
Example Calculation
Let’s say the paddle wheel does 500 J of work on the water, and we have 1000 g of water in the vessel. The specific heat capacity of water is 4.18 J/g°C. Plugging these values into our equation:
ΔT = Q / (mc) = 500 J / (1000 g × 4.18 J/g°C)
Calculating this gives:
ΔT = 500 / 4180 ≈ 0.1195°C
This means that the temperature of the water would rise by approximately 0.12°C due to the work done by the paddle wheel.
Final Thoughts
This example illustrates how mechanical work can be converted into thermal energy, leading to a temperature increase in a fluid. Understanding these principles is crucial in fields such as engineering and thermodynamics, where energy transfer plays a vital role in system design and efficiency.
